Electroconvective viscous fingering in a single polyelectrolyte fluid on a charge selective surface

When a low-viscosity fluid displaces into a higher-viscosity fluid, the liquid-liquid interface becomes unstable causing finger-like patterns. This viscous fingering instability has been widely observed in nature and engineering systems with two adjoined fluids. Here, we demonstrate a hitherto-unrealizable viscous fingering in a single fluid-solid interface. In a single polyelectrolyte fluid on a charge selective surface, selective ion rejection through the surface initiates i) stepwise ion concentration and viscosity gradient boundaries in the fluid and ii) electroconvective vortices on the surface. As the vortices grow, the viscosity gradient boundary pushes away from the surface, resulting viscous fingering. Comparable to conventional one with two fluids, i) a viscosity ratio (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document}M) governs the onset of this electroconvective viscous fingering, and ii) the boundary properties (finger velocity and rheological effects) - represented by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document}M, electric Rayleigh (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{Ra}}_{E}$$\end{document}RaE), Schmidt (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Sc}$$\end{document}Sc), and Deborah (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${De}$$\end{document}De) numbers - determine finger shapes (straight v.s. ramified, the onset length of fingering, and relative finger width). With controllable onset and shape, the mechanism of electroconvective viscous fingering offers new possibilities for manipulating ion transport and dendritic instability in electrochemical systems.


Supplementary Note 1. Fluid properties of polyelectrolyte solutions
Here, we describe how we obtain the viscosity of the polyelectrolyte solutions that is used as the viscosity of region (b) (  ) and the diffusivity of polyelectrolyte solution used for the effective diffusivity of the charged species.Since the PAA solution is known as a non-Newtonian fluid with yield stress and shear thinning properties 1,2 , we measured the zero-shear viscosity of the solutions, which is the viscosity of the fluid when it is effectively at rest, to represent the quiescent state of the region (b) (bulk fluid).To do so, through the rheometer test (ARES-G2, TA instrument Ltd., USA), we measured the shear stress with respect to the shear rate (Supplementary Fig. 11) and obtained the curve of the viscosity to the shear rate (Supplementary Fig. 12).At 0.1 wt% and 0.5 wt%, the PAA solution has a constant viscosity, so we simply choose the constant value of viscosity as the   (Supplementary Fig. 12a, b).In the case of 1.0-2.0wt% PAA and 0.5 wt% PQ-10 solutions, the viscosity increases as the shear rate decreases, and the viscosity peak appears at the low shear rate regime (Supplementary Fig. 12c-f).This peak represents the zero-shear limit of the viscoplastic fluid 3 , so we choose this peak value as the zero-shear viscosity and use it as   (the point that the viscosity decreases without fluctuation, red dotted point in Supplementary Fig. 12c-f).The zero-shear viscosity values in between experimental data (0.25, 0.75, 1.25, 1.75 wt%) were obtained through power regression in two different regimes (< 0.75 wt% and > 0.75 wt%, Supplementary Fig. 13).
Next, we explain how we obtain the diffusivities of the PAA solutions.As the PAA concentration increases in the aqueous solution, the distance between the PAA molecules becomes closer, resulting in an overlap between molecules 4 .The concentration that this molecular "overlap" occurs is called the overlap concentration ( * ), and it determines whether the solution is in dilute regime (  <  * , where the   is the polyelectrolyte concentration) or in semidilute regime (  >  * ) .The diffusivity of the PAA in the solution varies in these regimes and can be calculated by the following formulas 5    In PAA or PQ-10 solutions, the diffusivity of Na + /Cl -are presumed not changed in our experiment and scaling analysis.In the experiments with PQ-10, region (a) becomes concentrated with PQ-10 molecules.According to Ariel et al. 11 , if the size of the substance (e.g., ions) is much smaller than the intermolecular distance of the polymer in the solution, the substance barely interacts with the polymers, so the diffusivity of the substance is the same with that in the solvent without the polymer.
In this analogy, PQ-10's radius of gyration (~223 nm) is much larger than the ionic radius of Na + / Cl - (~0.1 nm), and the intermolecular distance is in the order of the radius of gyration even in the high concentration 12 .Therefore, we can assume that the diffusivity of Na + and Cl -in the PQ-10 solution would be constant in our experiment.In the experiments with PAA, since the PAA molecules migrate toward the anode, the depletion zone (region (a)) is assumed to be a pure water as most of PAA molecules are depleted.Therefore, we adopt the the diffusivity of Na + and Cl -in water, even the PAA's radius of gyration (~2.54 nm) is not much higher than the size of Na + /Cl -, resulting in   + = 1.33 × 10 −9 m 2 /s ,   − = 2.03 × 10 −9 m 2 /s , and   = 2 (1/  + + 1/  − ) ⁄ = 1.6071 × 10 −9 m 2 /s , where   + and   − is the diffusivity of Na + and Cl -, respectively.
Viscosities and effective diffusivities of PAA and PQ-10 solutions are summarized in Supplementary Table 1.
Here we describe the additional condition resulting from experiments with five polyelectrolytes in Supplementary Fig. 5: i) polyacrylic acid (PAA) as a weak anionic polyelectrolyte, ii) polyquaternium-10 (PQ-10) as a strong cationic polyelectrolyte, iii) sodium polystyrene sulfonate (NaPSS) as a strong anionic polyelectrolyte, iv) polyallylamine hydrochloride (PAH) as a weak cationic polyelectrolyte, and v) polyethylene oxide (PEO) as a neutral polymer.
Unlike the finger-like EC appearing in PAA and PQ-10 solutions, the conventional circular EC was observed in 1 wt% PAH and 1 wt% NaPSS solutions, both of which have a marginal viscosity ratio ( 11.9 and 1.66, respectively).ee may increase PAH NaPSS concentrations to achieve a higher viscosity ratio.However, this requires significantly high weight of these polyelectrolytes because their viscosity increase effect is relatively small (Supplementary Fig. 5a).If we increase the concentration to get high bulk viscosity (10 wt% of PAH NaPSS), the solution's conductance (9.8 23.16 mS for PAH NaPSS 10wt% solution, respectively) is too high to initiate strong ion depletion zone and EC (EC occurs when the concentration of NaCl solution at the membrane surface is sufficiently low for initiating ion depletion zone, and most of EC experiments are conducted under NaCl 0.1 M, where the conductance is 7.3 mS) 13,14 .Consequently, we can find one additional condition to generate EC fingering, i.e., the viscosity increase effect of a polyelectrolyte should be superior to the conductance increase effect according to its concentration.

Supplementary Note 3. The derivation of the finger velocity
Here we describe the governing equations used in our system and detailed derivation of finger velocity ( 0 ) in the text.To describe our system, we use two governing equations

Supplementary Note 4. pH/PAA concentration profiling
The pH change (1.5-6) and the PAA concentration (0-2 wt%) are represented by the gray value (Supplementary Fig. 4a).The correlation between pH and gray value is illustrated in the methods section in the manuscript.There are three regions according to the pH/PAA concentration profiles: i) diffusive ion enrichment zone on the top CEM with a linear increase of PAA concentration, ii) bulk region where the initial pH/concentration maintained (i.e., region (b) in Fig. 1c), and iii) ion depletion zone on the bottom CEM with EC and corresponding flat pH/concentration profiles (i.e., region (a) in Fig. 1c).During 8 sec operations at the applied voltage of 30 V in 0.5 wt% PAA solution, the diffusive enrichment zone expands only about 0.2 mm (which is well matched with the theoretical diffusion length, 0.1 mm ~√4  , see Supplementary Fig. 4b): whereas EC fingering grows much faster up to ~1 mm (Supplementary Fig. 4c-d).Interestingly, we can observe the EC-induced current hotspot between the fingers where the influx of the vortices occurs (white arrows in Supplementary Fig. 4a) 15 .
At this point, not only the depletion zone is suppressed, but also PAA is concentrated (Supplementary Fig. 4d).Accordingly, as the viscosity between the fingers increases, downward flows between the fingers are considerably suppressed whereas there are upward flows in the fingers.Such phenomenon is strengthened when EC are densely packed (e.g., 0.5 wt%, 30 V in Supplementary Video 1).
As described above, there are spatiotemporal variations of pH /PAA concentration in the ion enrichment /depletion zones during the developing EC fingers.However, these variations, except for change at the bulk-depletion region, are negligible for the following three reasons.First, the bulk region completely separates the ion enrichment and depletion zones until EC touches the enrichment zone after a considerable time (e.g., > 14 sec at the applied voltage of 30 V in 0.5 wt% PAA solution); at this merging moment, EC already determine its shape.Consequently, we can assume that the region (a) and (b) have constant viscosities.Also, while the viscosity between EC vortices increases as the PAA concentration increases, this is just one of the consequences of EC after it emerged.EC shape (circular or straight or ramified fingers) is still determined by the viscosity gradient of the ion depletion zone and the bulk region (i.e., viscosity ratio ).In addition, we estimated shear rates from the particle tracking images, and confirmed that it is mostly below 1 s -1 (Supplementary Fig. 2).Referring to Supplementary Fig. 12, the shear-thinning effect is not significant at below 1 s -1 shear rate, so we also can neglect the viscosity change by shear thinning of polyelectrolytes induced by EC vortices.Lastly, the pH variation in whole regions is between 1.5 to 4.5 (Supplementary Fig. 4).Therefore, we can expect the PAA molecules to be in a globular form and the radius of gyration is nearly constant everywhere 16 .
potential across the depletion zone (= ( −  ℎ )) - :  0 = 0.00089 Pa s is the viscosity of the solvent (water),   is the radius of gyration,   is the degree of polymerization, and   is the monomer concentration (see Supplementary Table4for molecular properties of the PAA we where   = 1.38064852 × 10 −23 m 2 kg s −2 K −1 is the Boltzmann's constant,  = 298 K is the absolute temperature (room temperature), * ) with equation (S. 2), the monomer concentration is linearly proportional to the PAA concentration (  ~ ), resulting   ~ −1   −0.5 .Therefore,   decreases as   increases (see Supplementary Table Momentum equation and Poisson's equation): where  is a density,  is velocity vector,  is a pressure,  is a viscosity,   is a charge density,  is an electrical potential,  is a permittivity and  ⃗ is an electric field.eithoutexternalpressure and inertia term in (S. 8), region (a) in the text (the depletion zone) can be described with the Stokes equation, which is0 =    2   +    2     .(S. 6)The subscript a indicates that the property is of the region (a) (so is subscript b in the region (b) below).Scaling with   ~ ,   ~ , and ~  , we can get where   is the velocity of EC in the region (a),   is the electrical potential across the depletion zone, and   is the size of EC.If we scale the finger velocity ( 0 =   /) as the   , we can get the  0 value by the following mathematical process:

Table 3 . Lists of symbols and dimensionless parameters
To scale Deborah number (1    ⁄ , where   is the relaxation time, and   is the processing time), the relaxation time represents as   =    ⁄ 17 , where the shear modulus of PAA solutions follows the scaling relation as ~    is the salt concentration,   is the Boltzmann's constant,  is the temperature, and   is the degree of polymerization).The viscosity   is represented to 0.0272  7.0918 in Supplementary Fig. 13 (  is the concentration of the PAA solution).After rearranging, we can present   ~ 0.917 with constant   and   in our experiments.The processing time is the exposure time that the viscosity gradient boundary is exposed affected to EC vortices, resulting   ~ /  ~   2  1 *In all symbols, the subscript a and b indicates that the property is of the region (a) and (b).*In all symbols, the tilde denotes dimensionless variables.